A general refinement of Jordan-type inequality

In this work, a general form of Jordan’s inequality: P2N(x)+aN+1(π2−4x2)N+1≤sinxx≤P2N(x)+1−∑n=0Nanπ2nπ2(N+1)(π2−4x2)N+1 is established, where x∈(0,π/2],P2N(x)=∑n=0Nan(π2−4x2)n,a0=2π,a1=1π3,an+1=2n+12(n+1)π2an−116n(n+1)π2an−1, and N≥0N≥0 is a natural number. The applications of the above result give the general improvement of the Yang Le inequality and a new infinite series (sinx)/x=∑n=0∞an(π2−4x2)n for 0<|x|≤π/20<|x|≤π/2.